91Ó°ÊÓ

Does a matrix have a unique row-echelon form? Illustrate your answer with examples. Is the reduced row-echelon form unique?

Consider the \(2 \times 2\) matrix \(\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]\) Perform the sequence of row operations. (a) Add (-1) times the second row to the first row. (b) Add 1 times the first row to the second row. (c) Add (-1) times the second row to the first row. (d) Multiply the first row by (-1) What happened to the original matrix? Describe, in general, how to interchange two rows of a matrix using only the second and third elementary row operations.

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) A system of one linear equation in two variables is always consistent. (b) A system of two linear equations in three variables is always consistent. (c) If a linear system is consistent, then it has an infinite number of solutions.

In your own words, describe the difference between a matrix in row-echelon form and a matrix in reduced row-echelon form.

Solve the system of equations by letting \(A=1 / x, B=1 / y,\) and \(C=1 / z\). $$\begin{array}{l} \frac{2}{x}+\frac{1}{y}-\frac{2}{z}=5 \\ \frac{3}{x}-\frac{4}{y}=-1 \\ \frac{2}{x}+\frac{1}{y}+\frac{3}{z}=0 \end{array}$$

Determine the value(s) of \(k\) such that the system of linear equations has the indicated number of solutions. Exactly one solution \(\begin{aligned} k x+2 k y+3 k z &=4 k \\ x+y+z &=0 \\ 2 x-y+z &=1 \end{aligned}\)

Find values of \(a, b,\) and \(c\) such that the system of linear equations has (a) exactly one solution, (b) an infinite number of solutions, and (c) no solution. \\[ \begin{aligned} x+5 y+z &=0 \\ x+6 y-z &=0 \\ 2 x+a y+b z &=c \end{aligned} \\]

Consider the system of linear equations in \(x\) and \(y\) \\[ \begin{array}{l} a x+b y=e \\ c x+d y=f \end{array} \\] Under what conditions will the system have exactly one solution?